A007334 -id:A007334 - cp's OEIS frontend

A089104 Duplicate of A007334.

Original entry on oeis.org

1, 2, 8, 50, 432, 4802, 65536, 1062882, 20000000, 428717762

Offset: 2

dead

A296366 Erroneous version of A007334.

Original entry on oeis.org

2, 8, 50, 442

Offset: 2

dead

Included in accordance with OEIS policy of including published but erroneous sequences to serve as pointers to the correct versions.

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence). See Entry M1883.

A058127 Triangle read by rows: T(j,k) is the number of acyclic functions from {1,...,j} to {1,...,k}. For n >= 1, a(n) = (k-j)k^(j-1), where k is such that C(k,2) < n <= C(k+1,2) and j = (n-1) mod C(k,2). Alternatively, table T(k,j) read by antidiagonals with k >= 1, 0 <= j <= k: T(k,j) = number of acyclic-function digraphs on k vertices with j vertices of outdegree 1 and (k-j) vertices of outdegree 0; T(k,j) = (k-j)k^(j-1).

Original entry on oeis.org

1, 1, 1, 1, 2, 3, 1, 3, 8, 16, 1, 4, 15, 50, 125, 1, 5, 24, 108, 432, 1296, 1, 6, 35, 196, 1029, 4802, 16807, 1, 7, 48, 320, 2048, 12288, 65536, 262144, 1, 8, 63, 486, 3645, 26244, 177147, 1062882, 4782969, 1, 9, 80, 700, 6000, 50000, 400000, 3000000, 20000000, 100000000

Offset: 1

Dennis P. Walsh, Nov 14 2000

An acyclic function f from domain D={1,...,j} to codomain C={1,...,k} is a function such that, for every subset A of D, f(A) does not equal A. Equivalently, an acyclic function f "eventually sends" under successive composition all elements of D to {j+1,...,k}. An acyclic-function digraph G is a labeled directed graph that satisfies (i) all vertices have outdegree 0 or 1; (ii) if vertex x has outdegree 0 and vertex y has outdegree 1, then x > y; (iii) G has no cycles and no loops. There is a one-to-one correspondence between acyclic functions from D to C and acyclic-function digraphs with j vertices of outdegree 1 and j-k vertices of outdegree 0.
n-th row of the triangle is the n-th iterate of "perform binomial transform operation" (bto) on current row to get next row, extracting the leftmost n terms for n-th row (i.e., all terms left of the zero). First row is (bto): [1, -1, 0, 0, 0, ...]. 5th row is 1, 4, 15, 50, 125, since (bto) performed 5 times iteratively on [1, -1, 0, 0, 0, ...] = 1, 4, 15, 50, 125, 0, -31, ... - Gary W. Adamson, Apr 30 2005
T(k,j) can be shown to be equal to the number of spanning trees of the complete graph on k vertices that contain a specific subtree with k-j-1 edges. - John L. Chiarelli, Oct 04 2016
T(k-1, j-1) is also the number of parking functions with j cars and k spots (see Theorem 2.2 in Kenyon and Yin). - Stefano Spezia, Apr 09 2021

			a(6) = T(3,2) = 3 because there are 3 acyclic functions from {1,2} to {1,2,3}: {(1,2),(2,3)}, {(1,3),(2,3)} and {(1,3),(2,1)}.
Triangle begins:
  1;
  1, 1;
  1, 2,  3;
  1, 3,  8,  16;
  1, 4, 15,  50,  125;
  1, 5, 24, 108,  432,  1296;
  1, 6, 35, 196, 1029,  4802, 16807;
  1, 7, 48, 320, 2048, 12288, 65536, 262144;
  ...

T. D. Noe, Rows n = 1..50 of triangle, flattened
Richard Kenyon and Mei Yin, Parking functions: From combinatorics to probability, arXiv:2103.17180 [math.CO] (2021).
Henri Mühle, Ballot-Noncrossing Partitions, Proceedings of the 31st Conference on Formal Power Series and Algebraic Combinatorics (Ljubljana), Séminaire Lotharingien de Combinatoire (2019) Vol. 82B, Article #7.
Jim Pitman and Richard P. Stanley, A polytope related to empirical distributions, plane trees, parking functions, and the associahedron, Discrete Comput. Geom. 27: 603-634 (2002).
D. P. Walsh, Notes on acyclic functions and their directed graphs

The sum of antidiagonals is A058128. The sequence b(n) = T(n, n-1) for n >= 1 is A000272, labeled trees on n nodes.

The sequence c(n) = T(n, n-2) for n >= 2 is A007334(n). The sequence d(n) = T(n, n-3) for n >= 3 is A089463(n-3,0). - Peter Luschny, Apr 22 2009

/* As triangle */ [[(n-k)*n^(k-1): k in [0..n-1]]: n in [1.. 10]]; // Vincenzo Librandi, Aug 11 2017

T := proc(n,k) (n-k)*n^(k-1) end; seq(print(seq(T(n,k),k=0..n-1)),n=1..9); # Peter Luschny, Jan 14 2009

t[n_, k_] := (n-k)*n^(k-1); Table[t[n, k], {n, 1, 10}, {k, 0, n-1}] // Flatten (* Jean-François Alcover, Dec 03 2013 *)

{T(n, k) = if( k<0 || k>n, 0, n==0, 1, (n-k) * n^(k-1))}; /* Michael Somos, Sep 20 2017 */

For fixed m = k-j, a(n) = T(k, j) = T(m+j, j) = m*(m+j)^(j-1). Exponential generating function g for T(m+j, j) = m*(m+j)^(j-1) is given by g(t) = exp(-m*W(-t)), where W denotes the principal branch of Lambert's W function. Lambert's W function satisfies W(t)*exp(W(t)) = t for t >= -exp(-1).

T(n, k) = Sum_{i=0..k} T(n-1, i) * binomial(k, i) if k < n. - Michael Somos, Sep 20 2017

a(32) corrected by T. D. Noe, Jan 25 2008

A232006 Triangular array read by rows: T(n,k) is the number of simple labeled graphs on vertex set {1,2,...,n} with exactly k components (all of which are trees) such that the labels {1,2,...,k} are all in distinct components (trees), n >= 0, 0 <= k <= n.

Original entry on oeis.org

1, 0, 1, 0, 1, 1, 0, 3, 2, 1, 0, 16, 8, 3, 1, 0, 125, 50, 15, 4, 1, 0, 1296, 432, 108, 24, 5, 1, 0, 16807, 4802, 1029, 196, 35, 6, 1, 0, 262144, 65536, 12288, 2048, 320, 48, 7, 1, 0, 4782969, 1062882, 177147, 26244, 3645, 486, 63, 8, 1, 0, 100000000, 20000000, 3000000, 400000, 50000, 6000, 700, 80, 9, 1

Offset: 0

Geoffrey Critzer, Nov 16 2013

Row sums = (n^n-n)/(n-1)^2 = A058128(n).

Column k without leading zeros is the k-th exponential (also called binomial) convolution of the sequence {A000272(n+1)} = {A232006(n+1, 1)}, for n >= 0, with e.g.f. LamberW(-x)/(-x), where LambertW is the principal branch of the Lambert W-function. This is also the row polynomial P(n, x) of the unsigned triangle A137452, evaluated at x = k. - Wolfdieter Lang, Apr 24 2023

			The triangle begins:
n\k  0         1        2       3      4     5    6   7  8 9 10 ...
0:   1
1:   0         1
2:   0         1        1
3:   0         3        2       1
4:   0        16        8       3      1
5:   0       125       50      15      4     1
6:   0      1296      432     108     24     5    1
7:   0     16807     4802    1029    196    35    6   1
8:   0    262144    65536   12288   2048   320   48   7  1
9:   0   4782969  1062882  177147  26244  3645  486  63  8 1
10:  0 100000000 20000000 3000000 400000 50000 6000 700 80 9  1
... Reformatted by _Wolfdieter Lang_, Apr 24 2023

R. P. Stanley, Enumerative Combinatorics, Cambridge, Vol. 2, 1999; see Proposition 5.3.2.

G. C. Greubel, Rows n=0..75 of triangle, flattened
Chad Casarotto, Graph Theory and Cayley's Formula, 2006
Alan D. Sokal, A remark on the enumeration of rooted labeled trees, arXiv:1910.14519 [math.CO], 2019.
Marc van Leeuwen, I am stuck with a combinatoric problem... Math Stackexchange, Answer May 14 2017.
Eric Weisstein's World of Mathematics, Lambert W-function
Wikipedia, Lambert W function

Cf. A058128, |A137452|.

Columns give A000007, A000272, A007334, A362354, A362355, A362356, ...

Prepend[Table[Table[k n^(n-k-1),{k,0,n}],{n,1,8}],{1}]//Grid

{T(n, k) = if( k<0 || k>n, 0, n^(n-k-1))}; /* Michael Somos, May 15 2017 */

T(n, k) = k*n^(n-k-1).
T(n, k) = Sum_{i=0..n-k} T(n-1, k-1+i)*C(n-k,i), T(0, 0) = 1, T(n, 0) = 0 when n >= 1.
From Wolfdieter Lang, Apr 24 2023: (Start)
E.g.f. for {T(n+k, k)}_{n>=0} is (LambertW(-x)/(-x))^k, for k >= 0.
T(n, k) = Sum_{m=0..n-k} |A137452(n-k, m)|*k^m, for n >= 0 and k = 0..n. That is, T(n, n) = 1, for n >= 0, and T(n, k) = Sum_{m=1..n-k} binomial(n-k-1, m-1)*(n-k)^(n-k-m)*k^m, for k = 0..n-1 and n >= k+1. (End)

A379611 Table read by rows: T(n, k) = (n + 1)^(n - 1) - (k - 1)*(n + 1)^(n - 2), by convention T(1, 0) = 1.

Original entry on oeis.org

2, 1, 1, 4, 3, 2, 20, 16, 12, 8, 150, 125, 100, 75, 50, 1512, 1296, 1080, 864, 648, 432, 19208, 16807, 14406, 12005, 9604, 7203, 4802, 294912, 262144, 229376, 196608, 163840, 131072, 98304, 65536, 5314410, 4782969, 4251528, 3720087, 3188646, 2657205, 2125764, 1594323, 1062882

Offset: 0

Peter Luschny, Dec 27 2024

			Triangle starts:
  [0]      2;
  [1]      1,      1;
  [2]      4,      3,      2;
  [3]     20,     16,     12,      8;
  [4]    150,    125,    100,     75,     50;
  [5]   1512,   1296,   1080,    864,    648,    432;
  [6]  19208,  16807,  14406,  12005,   9604,   7203,  4802;
  [7] 294912, 262144, 229376, 196608, 163840, 131072, 98304, 65536;

Steve Butler, Kimberly Hadaway, Victoria Lenius, Preston Martens, and Marshall Moats, Lucky cars and lucky spots in parking functions, arXiv:2412.07873 [math.CO], 2024. See p. 7, corollary 3.1.

Cf. A007334 (main diagonal), A374756, A375616, A379612 (column 0), A379613.

T := (n, k) -> ifelse(n=1 and k=0, 1, (n + 1)^(n - 1) - (k - 1)*(n + 1)^(n - 2)):

T[n_, k_] := T[n, k] = (n + 1)^(n - 1) - (k - 1)*(n + 1)^(n - 2); T[1, 0] := 1;
Flatten@ Table[T[n, k], {n, 0, 8}, {k, 0, n}] (* Michael De Vlieger, Dec 27 2024 *)

T(n, k) = (n + 1)^(n - 2)*(n - k + 2), if (n, k) != (1, 0).

T(n, k) = (1 - (k - 1)/(n + 1))*(n + 1)^(n - 1), if (n, k) != (1, 0).

A379613 a(n) = n^(n - 1) - 2*(n + 1)^(n - 2), by convention a(0) = 0.

Original entry on oeis.org

0, 0, 0, 1, 14, 193, 2974, 52113, 1034270, 23046721, 571282238, 15617863897, 467291386990, 15198954783153, 534222097472894, 20185726770649633, 816165851488045118, 35167910642711951617, 1609028732603454196606, 77912950297911241532841, 3981118415206568940420878

Offset: 0

Peter Luschny, Dec 27 2024

nonn

G. C. Greubel, Table of n, a(n) for n = 0..385
Steve Butler, Kimberly Hadaway, Victoria Lenius, Preston Martens, and Marshall Moats, Lucky cars and lucky spots in parking functions, arXiv:2412.07873 [math.CO], 2024. See p. 7.

Cf. A000169, A007334, A374756, A379611.

A379613:= func< n | n eq 0 select 0 else n^(n-1) -2*(n+1)^(n-2) >;
[A379613(n): n in [0..30]]; // G. C. Greubel, Mar 19 2025

a := n -> ifelse(n = 0, 0, n^(n-1) - 2*(n+1)^(n-2)): seq(a(n), n = 0..20);

{0}~Join~Table[n^(n - 1) - 2*(n + 1)^(n - 2), {n, 20}] (* Michael De Vlieger, Dec 27 2024 *)

def A379613(n): return 0 if n==0 else n^(n-1) -2*(n+1)^(n-2)
print([A379613(n) for n in range(31)]) # G. C. Greubel, Mar 19 2025

a(n) = A000169(n) - A007334(n+1) for n > 0. In the context of parking functions this is the difference between the main diagonals of A374756 and A379611. See corollary 3.1 and Table 2 in Butler et al.

E.g.f.: (1/(4*x))*((2*W(-x) + 2 - x)^2 - (4 - 12*x + x^2)), W(x) = Lambert W function. - G. C. Greubel, Mar 19 2025

A362353 Triangle read by rows: T(n,k) = (-1)^(n-k)binomial(n, k)(k+3)^n, for n >= 0, and k = 0,1, ..., n. Coefficients of certain Sidi polynomials.

Original entry on oeis.org

1, -3, 4, 9, -32, 25, -27, 192, -375, 216, 81, -1024, 3750, -5184, 2401, -243, 5120, -31250, 77760, -84035, 32768, 729, -24576, 234375, -933120, 1764735, -1572864, 531441, -2187, 114688, -1640625, 9797760, -28824005, 44040192, -33480783, 10000000, 6561, -524288, 10937500, -94058496, 403536070, -939524096, 1205308188, -800000000, 214358881

Offset: 0

Harlan J. Brothers and Wolfdieter Lang, Apr 27 2023

This is the member N = 2 of a family of signed triangles with row sums n! = A000142(n): T(N; n, k) = (-1)^(n-k)*binomial(n, k)*(k + N + 1)^n, for integer N, n >= 0 and k = 0, 1, ..., n. The row polynomials PS(N; n, z) = Sum_{k=0..n} T(N; n, k)*z^k = ((-1)^n/z^N)*D_{n,N+1,n}(z) in [Sidi 1980].
For N = -1, 0 and 1 see A258773(n, k), A075513(n+1, k) and (-1)^(n-k) * A154715(n, k), respectively.
The column sequences, for k = 0, 1, ..., 6 and n >= k, are A141413(n+2), (-1)^(n+1)*A018215(n) = 4*(-1)^(n+1)*A002697(n), 5^2*(-1)^n*A081135(n), (-1)^(n+1)*A128964(n-1) = 6^3*(-1)^(n+1)*A081144(n), 7^4*(-1)^n*A139641(n-4), 2^15*(-1)^(n+1)*A173155(n-5), 3^12*(-1)^n*A173191(n-6), respectively.
The e.g.f. of the triangle (see below) needs the exponential convolution (LambertW(-z)/(-z))^2 = Sum_{n>=0} c(2; n)*z^n/n!, where c(2; n) = Sum_{m=0..n} |A137352(n+1, m)|*2^m = A007334(n+2).
The row sums give n! = A000142(n).

			The triangle T begins:
n\k    0       1        2         3         4          5          6         7
0:     1
1:    -3       4
2:     9     -32       25
3:   -27     192     -375       216
4:    81   -1024     3750     -5184      2401
5:  -243    5120   -31250     77760    -84035      32768
6:   729  -24576   234375   -933120   1764735   -1572864     531441
7: -2187  114688 -1640625   9797760 -28824005   44040192  -33480783  10000000
...
n = 8:  6561 -524288 10937500 -94058496 403536070 -939524096 1205308188 -800000000 2143588,
n = 9: -19683 2359296 -70312500 846526464 -5084554482 16911433728 -32543321076 36000000000 -21221529219 5159780352.

Paolo Xausa, Table of n, a(n) for n = 0..5049 (rows 0..100 of the triangle, flattened)
Wolfdieter Lang, On a Certain Family of Sidi Polynomials, May 2023.
Avram Sidi, Numerical Quadrature and Nonlinear Sequence Transformations; Unified Rules for Efficient Computation of Integrals with Algebraic and Logarithmic Endpoint Singularities, Math. Comp., 35 (1980), 851-874.

Cf. A000142 (row sums), A075513, A154715, A258773.

Columns k = 0..6 involve (see above): A002697, A007334, A018215, A081135, A081144, A128964, A137352, A139641, A141413, A173155, A173191.

A362353row[n_]:=Table[(-1)^(n-k)Binomial[n,k](k+3)^n,{k,0,n}];Array[A362353row,10,0] (* Paolo Xausa, Jul 30 2023 *)

T(n, k) = (-1)^(n-k)*binomial(n, k)*(k + 3)^n, for n >= 0, k = 0, 1, ..., n.
O.g.f. of column k: (x*(k + 3))^k/(1 - (k + 3)*x)^(k+1), for k >= 0.
E.g.f. of column k: exp(-(k + 3)*x)*((k + 3)*x)^k/k!, for k >= 0.
E.g.f. of the triangle, that is, the e.g.f. of its row polynomials {PS(2;n,y)}_{n>=0}): ES(2;y,x) = exp(-3*x)*(1/3)*(d/dz)(W(-z)/(-z))^2, after replacing z by x*y*exp(-x), where W is the Lambert W-function for the principal branch. This becomes ES(2;y,x) = exp(-3*x)*exp(3*(-W(-z)))/(1 - (-W(-z)), with z = x*y*exp(-x).

a(41)-a(44) from Paolo Xausa, Jul 31 2023

A385085 a(n) = 2 * (3*n+2)^(n-1).

Original entry on oeis.org

1, 2, 16, 242, 5488, 167042, 6400000, 296071778, 16063620352, 1000492825922, 70368744177664, 5517094707031250, 477144100447105024, 45126980600732372162, 4633559988356427808768, 513333972375334818668738, 61035156250000000000000000, 7752538100237033690795744642

Offset: 0

Seiichi Manyama, Jun 17 2025

Eric Weisstein's World of Mathematics, Lambert W-Function.

Cf. A007334, A385086.

Cf. A052752.

my(N=20, x='x+O('x^N)); Vec(serlaplace(exp(-2/3*lambertw(-3*x))))

E.g.f.: exp(-2/3 * LambertW(-3*x)).
E.g.f.: B(x)^2, where B(x) is the e.g.f. of A052752.
E.g.f. A(x) satisfies:
(1) A(x) = exp(2*x*A(x)^(3/2)).
(2) A(x) = 1/A(-x*A(x)^3).

A385086 a(n) = 2 * (5*n+2)^(n-1).

Original entry on oeis.org

1, 2, 24, 578, 21296, 1062882, 67108864, 5131452818, 461078666496, 47622573323522, 5559811767271424, 724066662913782498, 104073121367674187776, 16365437809265714289122, 2794811034494209364066304, 515110198093444174897047218, 101914923171285428527995355136

Offset: 0

Seiichi Manyama, Jun 17 2025

Eric Weisstein's World of Mathematics, Lambert W-Function.

Cf. A007334, A385085.

Cf. A052782.

my(N=20, x='x+O('x^N)); Vec(serlaplace(exp(-2/5*lambertw(-5*x))))

E.g.f.: exp(-2/5 * LambertW(-5*x)).
E.g.f.: B(x)^2, where B(x) is the e.g.f. of A052782.
E.g.f. A(x) satisfies:
(1) A(x) = exp(2*x*A(x)^(5/2)).
(2) A(x) = 1/A(-x*A(x)^5).

A338280 Triangle T read by rows: T(n, k) = k*n^(n-k-1) with 0 < k < n.

Original entry on oeis.org

1, 3, 2, 16, 8, 3, 125, 50, 15, 4, 1296, 432, 108, 24, 5, 16807, 4802, 1029, 196, 35, 6, 262144, 65536, 12288, 2048, 320, 48, 7, 4782969, 1062882, 177147, 26244, 3645, 486, 63, 8, 100000000, 20000000, 3000000, 400000, 50000, 6000, 700, 80, 9, 2357947691, 428717762, 58461513, 7086244, 805255, 87846, 9317, 968, 99, 10

Offset: 2

Stefano Spezia, Oct 20 2020

T(n, k) is the number of forests of n - k edges that connect every other labeled vertex to one of the k roots (see Section 3 in Wästlund).

Alfred Rényi, Some remarks on the theory of trees. MTA Mat. Kut. Inst. Kozl. (Publ. math. Inst. Hungar. Acad. Sci) 4 (1959), 73-85.

Arthur Cayley, A theorem on trees, Quart. J. Pure Appl. Math. 23: 376-378 (1889). Also in The collected mathematical papers of Arthur Cayley vol 13.
John Riordan, Forests of labeled trees, Journal of Combinatorial Theory 5 (1968), 93-103.
Lajos Takács, On Cayley’s Formula for Counting Forests, Journal of Combinatorial Theory Series A 53, 321-323 (1990). See Equation 1.
Johan Wästlund, Padlock Solitaire: A martingale trick for combinatorial enumeration, arXiv:2008.13017 [math.CO], 2020. See Section 3.

Cf. A000027 (diagonal), A000169, A000272 (1st column), A000312, A007334 (2nd column), A023811 (row sums), A034941, A072590, A075363, A210725.

Table[k*n^(n-k-1),{n,2,11},{k,1,n-1}]//Flatten

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A089104 Duplicate of A007334.

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A296366 Erroneous version of A007334.

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A232006 Triangular array read by rows: T(n,k) is the number of simple labeled graphs on vertex set {1,2,...,n} with exactly k components (all of which are trees) such that the labels {1,2,...,k} are all in distinct components (trees), n >= 0, 0 <= k <= n.

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A379611 Table read by rows: T(n, k) = (n + 1)^(n - 1) - (k - 1)*(n + 1)^(n - 2), by convention T(1, 0) = 1.

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A379613 a(n) = n^(n - 1) - 2*(n + 1)^(n - 2), by convention a(0) = 0.

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A362353 Triangle read by rows: T(n,k) = (-1)^(n-k)*binomial(n, k)*(k+3)^n, for n >= 0, and k = 0,1, ..., n. Coefficients of certain Sidi polynomials.

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A385085 a(n) = 2 * (3*n+2)^(n-1).

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A362353 Triangle read by rows: T(n,k) = (-1)^(n-k)binomial(n, k)(k+3)^n, for n >= 0, and k = 0,1, ..., n. Coefficients of certain Sidi polynomials.